This is the final preprint version of a paper which appeared in Algebra Universalis, 68 (2012) 17-37. The published version is accessible to subscribers at http://dx.doi.org/10.1007/s00012-012-0191-2 . Isotone maps on lattices G. M. Bergman and G. Gratzer¨ Abstract. Let L = (Li j i 2 I) be a family of lattices in a nontrivial lattice vari- ety V, and let 'i : Li ! M, for i 2 I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps 'i can be extended to an isotone map ': L ! M, where L = FreeV L is the free product of the Li in V. This was known for L = V, the variety of all lattices. The above free product L can be viewed as the free lattice in V on the partial lattice P formed by the disjoint union of the Li. The analog of the above result does not, however, hold for the free lattice L on an arbitrary partial lattice P . We show that the only codomain lattices M for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices P that are not-quite-disjoint unions of lattices. We also obtain some results similar to our main one, but with the relationship lat- tices : orders replaced either by semilattices : orders or by lattices : semilattices. Some open questions are noted. 1. Introduction By Yu. I. Sorkin [15, Theorem 3], if L = (Li j i 2 I) is a family of lattices and 'i : Li ! M are isotone maps of the lattices Li into a lattice M, then there exists an isotone map ' from the free product Free L of the Li to M that extends all the 'i. (There are some difficulties with Sorkin's original proof; but a simple, correct proof is given by G. Gr¨atzer,H. Lakser and C. R. Platt in [11, x4].) 2010 Mathematics Subject Classification: Primary: 06B25. Secondary: 06B20, 06B23. Key words and phrases: Free product of lattices; varieties, prevarieties and quasivarieties of lattices; isotone map; free lattice on a partial lattice; semilattice. 2 G. M. Bergman and G. Gr¨atzer Algebra univers. Our main result, proved in Section 2, is a generalization of this fact, with Free L replaced by FreeV L, the free product of the Li in any nontrivial va- riety V of lattices containing them | though not necessarily containing M. (In Section 3, we explore some variants of our proof of this result.) We may regard FreeV L as the free lattice in V on the partial lattice P given by the disjoint union of the Li. Does the analog of the above result hold for more general partial lattices P and their free lattices L? In Section 4 we find that the lattices M such that this statement holds for all partial lattices P are the complete lattices. On the other hand, we describe in Section 5 a class of partial lattices P , related to but distinct from the class considered in Section 2, for which the full analog of the result of that section holds. Since semilattices lie between orders and lattices, it is plausible that state- ments similar to our main result should hold, either with \lattice" weakened to \semilattice", or with \lattice" unchanged but \isotone map" strengthened to \semilattice homomorphism". In Section 6 we shall find that the former statement is easy to prove. In that section and Section 7, we obtain sev- eral approximations to the latter statement; we do not know whether the full statement holds. The reader familiar with the concepts of quasivariety and prevariety will find that the proofs given in this note for varieties of lattices in fact work for those more general classes. However, varieties are not sufficient for the result of Section 7, so we develop that in terms of prevarieties. In Section 8 we note some open questions. For general definitions and results in lattice theory, see [8] or [9]. For another context in which isotone maps among lattices have appeared in the literature|in this case, isotone sections of lattice homomorphisms|see R. Freese, J. Jeˇzek,and J. B. Nation [6, pp. 99{107, 133] and R. Freese [5]. We are indebted to R. Freese for pointing us to those works, and for the related observation following Corollary 4 below; and to A. V. Kravchenko for an extensive correspondence regarding Sorkin's original argument. 2. Extending isotone maps to free product lattices Let V be a nontrivial lattice variety, that is, a variety V of lattices having a member with more than one element. Let L = (Li j i 2 I) be a fam- ily of lattices in V, and L = FreeV L their free product in V. Finally, let ('i : Li ! M j i 2 I) be a family of isotone maps into a lattice M, not as- sumed to lie in V. To show that the 'i have a common extension to L, it suffices, by the 0 universal property of L, to find some L 2 V such that each map 'i factors 0 Li ! L ! M, where the first map is a lattice homomorphism, and the second an isotone map not depending on i. So let us, for now, forget free products, and obtain such a lattice L0. Vol. 00, XX Isotone maps on lattices 3 We first note (as remarked in [11, last paragraph] for V = L) that this is easy if M has a least element, or more generally, if its subsets 'i(Li) have a common lower bound e 2 M. In that case, we begin by enlarging all the Li to lattices L¯i = feig + Li, where ei is a new least element, and extend the 'i to maps 0 Q '¯i : L¯i ! M mapping ei to e. Now let L be the sublattice of (L¯i j i 2 I) consisting of the elements x = (xi j i 2 I) such that xi = ei for all but 0 finitely many i; and let us map each Li into L by the homomorphism sending x 2 Li to the element having i-component x, and j-component ej for all j 6= i. We now map L0 to M using the isotone map given by W 0 (x) = (' ¯i(xi) j i 2 I ) for x = (xi j i 2 I) 2 L . (1) This infinite join is defined because all but finitely many of the joinands are 0 e; and it is easy to verify that for each i, the composite map Li ! L ! M is the given isotone map 'i, as required. If, rather, the 'i(Li) have a common upper bound, the dual construction is, of course, available. In the absence of either sort of bound, we shall follow the same pattern of adjoining to the Li elements ei with a common image e in M (this time an arbitrary element of that lattice); but that construction takes a bit more work, as does the one analogous to the definition (1) of the isotone map : L0 ! M. The first of these steps is carried out in the following lemma (where L corre- sponds to the above Li). Lemma 1. Let ': L ! M be any isotone map of lattices, and e any ele- ment of M. Then there exists a lattice extension L¯ of L, and an isotone map '¯: L¯ ! M extending ', such that e 2 '¯(L¯). Moreover, L¯ can be taken to lie in any nontrivial lattice variety V containing L. Proof. Let L¯ = L × C2 × C2, where C2 is the 2-element lattice f0; 1g, and embed L in L¯ by x 7! (x; 0; 1). Define' ¯: L¯ ! M by 8 > '¯(x; 0; 1) = '(x); > > <> '¯(x; 1; 0) = e; (2) > '¯(x; 0; 0) = '(x) ^ e; > > :> '¯(x; 1; 1) = '(x) _ e: It is easy to check that' ¯ is isotone, and it clearly has e in its range. Since C2 = f0; 1g belongs to every nontrivial variety of lattices, L¯ will belong to any nontrivial variety V containing L. The next lemma gives the construction we will use to weld our I-tuple of isotone maps into one map. Lemma 2. Let M be a lattice, e any element of M, and I a nonempty set. Let M 0 be the sublattice of M I consisting of those elements f such that f(i) = e 4 G. M. Bergman and G. Gr¨atzer Algebra univers. for all but finitely many i 2 I. Then there exists a map : M 0 ! M such that is isotone (3) and For every i 2 I, and every f 2 M 0 satisfying f(j) = e for all j 6= i, (4) we have (f) = f(i). Proof. For f 2 M 0, define 8 <V( f(i) j i 2 I ) if f(i) ≤ e for all i 2 I, (f) = (5) W : ( f(i) j i 2 I; f(i) e ) otherwise. These meets and joins are defined because for each f 2 M 0, there are only finitely many distinct values f(i). It is easy to see that satisfies (4). To obtain (3), observe that For f ≤ g in M 0, we have fi j f(i) 6≤ eg ⊆ fi j g(i) 6≤ eg. (6) Hence given f ≤ g, there are three possibilities: Either the definitions of (f) and (g) both fall under the first case of (5), or they both fall under the second, or that of (f) falls under the first and that of (g) under the second.